|
Henri Poincaré : A decisive contribution to Special Relativity The short story by Jacques. Fric (June 2003)
Acknowledgements: This article relies on a paper published by Jules Leveugle [ ref 7] and a following paper published by Christian Marchal : Poincaré: une contribution decisive à la Relativité Introduction
In April 1994 « la Jaune et la Rouge » published a survey from Jules Leveugle (Ecole Polytechnique) named «Poincaré et la Relativité » (Réf 7). Evidences of the major contribution of Henri Poincaré to the Special Relativity theory are presented in this survey. According to the success encountered by this article, and the questions about it, Christian Marchal published an additional paper in order to answer to the questions. H. Poincaré is very well known as a famous mathematician, even though he never taught mathematics but physics (electromagnetism) as a professor at the “Ecole Polytechnique”. His work on Relativity is less known. The purpose of this paper is to demonstrate that he set up all the basic concepts of Special Relativity, several years before Einstein did in his the famous paper (1905: Annalen der Physik vol XVII 1905 p 891-921: ref 6). We will review briefly the controversy between Lorentz and H. Poincaré about absolute space time (1898-1905). We will compare the very different approaches of Poincaré (formal) and Einstein (physical), on Relativity (illustrated by the demonstration, within the theory, of the Lorentz equations by each of them) . We will end on some questions: Why the work of Poincaré is so ignored, why Einstein did not mentioned any references at all, in his 1905 fundamental article, acting as he had invented everything, even though, at least the Relativity principle as well as the method for synchronizing clocks are borrowed from Poincaré published papers (1898-1902). This should explain why Einstein was not awarded the Nobel Prize for the Relativity, but for the photo-electric effect! For a detailed bibliography of H. Poincaré see http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html …. Lorentz – Poincaré Controversy (1898-1905) At the end of the 19th century, when the detection (at the second order) by the Michelson Morley experience of the absolute motion “in the ether” of the Earth failed, the science was under trouble. In 1895 Lorentz suggested that moving bodies should experience a “physical” contraction, compute what it would be in accordance to the Michelson Morley experience and set up a theory on this basis. In 1899, Henri Poincaré, who was teaching electromagnetism at the “Ecole Polytechnique” wrote in one of his course [ref: “Electricité et Optique”published by Carré et Nadaud, 1901, p 536 ] about this hypothesis: “ I am not satisfied with the explanation ( physical contraction) of the negative result of the Michelson experiment by the Lorentz theory, I would say that the laws of optics are only depending on the relative motion of the involved bodies”. It was the beginning of the ( friendly) controversy with Lorentz , mainly about the concept of absolute Space and absolute time. In his book “La science et l’hypothèse” (1902), Poincaré devoted a full chapter to the relativity principle: “There is no absolute uniform motion, no physical experience can therefore detect any inertial motion (no force felt), there is no absolute time, saying that two events have the same duration is conventional, as well as saying they are simultaneous is purely conventional as they occur in different places.” One can still keep in mind the concept of ether, if it helps for thinking, but it is a unphysical concept, it is a metaphysical concept. He defines the way to synchronise all the clocks of an inertial frame, by using light signals (1900: La théorie de Lorentz et le principe de réaction, published J.Bosscha]. In 1904 at the St Louis conference, he proposes to add the Relativity principle to the five classical “universal” principles of the physics. He emphasised that the Lorentz contraction was an “ad hoc” hypothesis, just made for adjusting the theory to the experience. Lorentz reported this point of view of Poincaré, in his theory of electromagnetism (1904-Ref [1]). “Poincaré has objected to the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelson’s negative result, the introduction of a new hypothesis has been required, and that the same necessity may occur each time new facts will be brought to light. Surely this course of inventing special hypotheses for each new experimental result is somewhat artificial. It would be more satisfactory if it were possible to show by means of certain assumptions and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system…..” He points out that the “form” of the Lorentz formulae can be demonstrated from the Relativity principle alone, ( see annex 1) and therefore are implied by this principle alone (with a parameter to specify, related to “c” for the Special relativity), only one hypothesis is required, instead of the local time and the associated three new hypothesis of the article of Lorentz ( 1904). At last but not at least, In 1900, he noticed that the recoil of a radiation, of energy E, is m = E/c² [ ref oeuvres de Poincaré ,op.cit.t IX p 471] which is nothing else that the famous E = mc². We can see that in 1905 Poincaré, in different articles, had set up all the basic concepts of the special Relativity even though he did not summarize all of them in the framework of a formal theory as Einstein did.
Special Relativity : Poincaré versus Einstein On one hand, Poincaré as a mathematician had a clear formal approach of the Relativity principle. Whether we have a look at his demonstration (annex 1), we see that he states first that inertial frames are homogenous allowing to express each coordinate (x,t) of one frame as a linear function of the coordinates of the other frame (x’,t’) and vice versa , he had previously rotated one axis by 180 ° in order to have a full symmetrical situation. This results in four relations with eight unknown parameters. The symmetry of the situation reduces them to four and using relations leaves the equations with only one undetermined parameter, function of the relative speed. The form of this last parameter can be computed taking into account that into a group, by definition, operation by two elements is an element of the group. He considered three inertial frames (1), (2), (3) and applied the equations according (1) ->(3) = (1)->(2)->(3): he got: . x'=(x-Vt) (1-V²/K)-1/2 ; y'=y ; z'=z t'=[(t-(Vx/K)] (1-V²/K)-1/2 (1) Note that in this demonstration, he never uses the (second) postulate ( constancy of the speed of light). So it is more general. Where V is the relative velocity of the two frames and K is a parameter having the dimension of a square of a speed which, is the maximum possible value of V (to keep the equation real) . Whether K = c², we recognize the Lorentz equations, whether K = infinity, it is Newton. The interesting thing is that each value of K (continuous parameter) defines a continuous family (with a group structure) of inertial frames. The least we can say is that this demonstration is very simple. Poincaré understood that such Relativity principle reflects symmetry properties of the physical laws (prefiguring Noether theorem: 1915): invariance of physical laws under a group of transformations : the Poincaré group, [ Ref 3] including three spatial rotations, three hyperbolic rotations (boosts) and four translations, conferring to the Minkowski space ( The special Relativity spacetime), the maximum possible number of symmetries : n( n+1)/2 = 10. This is obviously, the modern approach. . On the other hand, Einstein, more intuitive and sticking to physical principles, first states the two postulates: Relativity principle ( borrowed to Poincaré) and constancy of light speed in all Galilean frames. Then he studies the methods for synchronizing clocks in a frame ( borrowed to Poincaré) , allowing also to define space milestones ( with the help of the constant light speed). Then, after defining exactly the terminology, he considers a moving frame and writes the relations between the coordinates of the two frames by considering the clock synchronisation equation in the two different frames derived from the light speed constancy. See equation (5) in annex 2. In this step by step demonstration, we discover the expected result only at the end.. The last part is quite classical. The paper of Einstein [ ref 6] is a more extensive treaty about Relativity, as in addition to the demonstration of the Lorentz formula, he considers also the cinematic, and the dynamic of the electron, it is a first synthesis about the theory of special relativity and its implications. ( E = mc² will come later in a second paper [Annalen der Physic, 1905 vol XVIII, p 639-641])
That’s the facts. So, why the work of Poincaré in Special Relativity is so ignored ? One point is that Poincaré did not realize the revolution implied by this new theory, as he had many others topics of interest especially in mathematics. His formal (maybe too speculative) approach did not reflect the “physical reality” implied by the theory. His position on the ether remains quite ambiguous, as he did not reject definitely the concept as Einstein did (at the beginning, as later Einstein had a more ambiguous position too, ether being a “hard to kill “concept). In some following papers, he expresses some doubts about the universality of the Relativity principle. In addition, he never published a so extensive treaty than Einstein. Einstein approach more physical, more practical, was closer to the physical reality, and may be seen more striking than the Poincaré approach. In 1905 Poincaré, who was in his fifty’s, was known as a shy person (For instance, he did most of the work for the Fuchsian functions, but let the paternity to Fuchs). He was recognised as a famous mathematician, his comfortable situation and his reputation can explain his cautious and tedious attitude. Soon later, he got seriously ill. He got a cancer in 1909 and died in 1912. Einstein was young, bold, unknown at that time, and therefore took over the whole thing. Einstein has always denied to have known Poincaré publications. It’s hard to believe as his friends Maurice Solovine and Carl Seelig, report Einstein had read the the Poincaré book ” La Science et l'hypothèse” (no absolute time, no absolute space, no ether ... ) around 1902-1904. This book was commented at their reading commitee « Académie Olympia » during several weeks (ref. 8, pages 129 et 139 ; ref. 9, page VIII and ref. 17, page 30) His position at the Swiss office patent in charge of “electromagnetism” implied that part of his job was to read and summarize the main publications on this topic ( he summarized several papers from the French “Academie des sciences”). At the end of his life, Einstein wrote in 1955 in a letter to Carl Seelig: «There is no doubt, if we look back to the development of the Relativity theory, special Relativity was about to be discovered in 1905. Lorentz already noticed that the transformations ( named Lorentz transformations) were essential in the Maxwell theory and Poincaré had gone even further. At that time I only knew Lorentz work of 1895, but I knew neither Lorentz nor Poincaré further work. This why I can say that my work of 1905 was independent » (ref 8, page 11). Anyway, Einstein had a different approach, he popularised the Special Relativity, and developed “ alone” the General Relativity ( even though Hilbert derived the field equation of GR a few days before Einstein, but by using Einstein work so the equation was rightly attributed to Einstein) . When the Nobel Committee decided to award Einstein the price, Lorentz, who was a member, objected that it was Poincaré who had found the full group of transformations of the SR, and it would be unfair not to associate him. But as Poincaré was dead at that time, Einstein was awarded the Nobel price for his work on the photo electric effect: That’s diplomacy…
. Références
1 Lorentz H.A. Electromagnetic phenomena in a system moving with any velocity less than that of light. Proc.Royal Acad. Amsterdam, 6, page 809, 1904. 2 Poincaré H. La Science et l'hypothèse. Edition Flammarion, Paris, 1902. 3 Poincaré H. Sur la dynamique de l'électron. Comptes rendus Acad. Sci. Paris, 140, pages 1504-1508, 5 Juin 1905. 4 Poincaré H. La mesure du temps. Revue de métaphysique et de morale. 6, pages 371384, 1898. 5 Poincaré H. Sur la dynamique de l'électron. Rendiconti del Circolo Matematico di Palermo, 21, pages 129-175, reçu le 23 Juillet 1905, published in January 1906. 6 Einstein A. Zur Elektrodynamik der bewegten Körper. Annalen der Physik, 17, pages 891-921, reçu le 30 Juin 1905, publié le 26 Septembre 1905. 7 Leveugle J. Poincaré et la relativité. La Jaune et la Rouge, pages 3 1-5 1, Avril 1994. 8 Miller A.I. Albert Einstein's Special Theory of Relativity. Ed. Addison-Wesley Publishing Company Inc. Reading Mass., 1981. 9 Solovine M. Lettres à Maurice Solovine. Ed. Gauthier-Villars, Paris, 1956. 10 Lorentz H.A. Deux mémoires de Henri Poincaré dans la Physique mathématique. Acta Mathematica, 38, pages 293-308, 1921. 1 1 Poincaré H. L'état actuel et l'avenir de la physique mathématique. Bulletin des Sciences Mathématiques, 28, 2° série (réorganisé 39-1), pages 302-324, 1904. 12 Tonnelat M.A. Histoire du principe de relativité. Ed. Flammarion, Paris, 1971. 13 Ginzburg V.L. On the theory of relativity. Ed. Nauka, Moscow, 1979. 14 Bol'shaia Sovetskaia Entsiklopedia. Great Soviet Encyclopedia-A translation of the third edition. Volume 18, Macmillan Inc., New-York, Collier Macmillan Publishers. Relativity, Theory of, page 653, 1974. 15 Pauli W., Kottler F. Encyclopädie der mathematsichen Wissenchaften.Leipzig Verlag und Druck von B G Teubner. Relativitätstheorie V-2, pages 545-546 (1904-1922)Gravitation und Relativitätstheorie VI-2-2, page 171 (1922-1934). 16 Logunov A. A. On the articles by Henri Poincaré: « On the dynamics of the electron » Publishing Dept of the Joint Institute for Nuclear Research, Dubna, 1995. Sur les articles de Henri Poincaré : « Sur la dynamique de l'électron ». Le texte fondateur de la Relativité en langue scientifique moderne. Publication ONERA 2000-1, pages 1-48, 2000. 1 7 Merleau-Ponty J. Einstein. Ed Flammarion ISBN, page 30, 1993. 1 8 Einstein A.Beiblâtter zu der Annalen der Physik. 29, N' 18, pages 952-953, 1905. 1 9 Einstein A. L'éther et la théorie de la relativité. Conférence faîte à Leyde (Pays-Bas) le 5 Mai 1920. Traduction en français par Maurice Solovine et M.A. Tonnelat dans: Albert Einstein, Réflexions sur l'électrodynamique, l'éther , la géométrie et la relativité. Collection « Discours de la méthode », nouvelle édition, Gauthier-Villars éd. 55 Quai des Grands Augustins, Paris 6è, page 74,1972. 20 Darrigol 0. Henri Poincaré 's criticism of Fin de Siècle electrodynamics Studies in History and Philosophy of modem Physics, pages 1-4, April 1995.
References 3, 5 et 10 appear also in "Oeuvres de Henri Poincaré", respectively tome 9, pages 489-493 ; tome 9, pages 494-550 et tome 11, page 247-261; Gauthier-Villars éditeur, Paris, 1956.
The Lorentz Transformation ( Poincaré) The basic idea is that coordinates transformations form a group. We will derive it, by using the symmetries implied by the Relativity principle alone. A- First general considerations
Let O'x' be one inertial frame sliding on Ox, an other inertial frame at a constant speed V. ___________________________________________O'___________________________> x' ______________________O______OO'= V_____________________________________> x
In order to have a full symmetrical configuration, let’s rotate O’x’ by 180°, as shown below. x'<_________________________________________O'________________ ________________________O_____________________________________> x
Homogeneity would imply a linear transformation. If we set t = t' = 0 in Ox and O'x’ frames, when O and O' coincide : transformations (x, t) ® (x', t') and (x', t')® (x, t) will be as follow, with eight constants A to D'
(4) x' = Ax + Bt t' = Cx + Dt x = A'x' + B't' t = C'x' + D't' Relativity principle and symmetry imply: (5) A = A' B = B' C = C' D = D' In addition, for O' we have x' = 0 and x = Vt, therefore x' = Ax + Bt implies AV + B = 0, as well as x = Ax' + Bt' and t = Cx' + Dt' implies B = DV, and therefore D = -A. Finally, consistency implies: (6) x = Ax' + Bt' = A(Ax + Bt) + B(Cx + Dt) = (D' + CDV)X t = Cx' + Dt' = C(Ax + Bt) + D(Cx + Dt) ='(D' + CDV) t, therefore D^2 + CDV = 1, that is: C = (1 – D^2) / DV. The transformation (x, t) -> (x', t') becomes
(7) x' = -Dx + DVt, t' = [(l – D^2) / DV]x + Dt B- The only unknown parameter, D, is obviously a function of the speed V. It will be determined by considering that the “product” of two transformations is a transformation of the group and computing it in two different ways : Now we resume O'x' initial position and let consider three axes Ox, O'x' et O"x" oriented in the same way .
__O"____________________________>x" OO"=V"t, O'O" = V't ____________________________________O'__________________________________>x' OO' = Vt __________________________O_____________________________________________>x
The relation (7) becomes with opposite sign for x'
(8) x' = D(x - Vt) , t'= [(l – D^2) / DV]x + Dt
If we do the same for D' with V' and D" with V "(This new D' is not the same than the one used in (4)-(5), no longer used after (5)) we get :
(9) x"= D'(x'- V't'); t" = [(l - D'^2)/D'V']x' + D't' (10) x"= D"(x - V"t); t" = [(l - D"^2)/ D"V"]x + D"t
Plugging x' and t' from (8) into (9), we get an other expression of (10)
(11) x"={DD'+[D'V'(D^2 - 1) / DV] }x - DD'(V + V')t t"={[(D - DD'^2) / D'V'] + [(D'- D^2. D') / DV]} x + { DD'+ [DV(D'^2 - 1) /D'V']} t
Identifying (10) and (11) leads to the four equations described below: (12) D " = DD' + [D'V'(D^2 -1) / DV] (13) D"V "= DD'(V + V') (14) (1 - D"^2 ) / D"V"= [(D - DD'^2) /D'V'] + [(D'- D^2.D') / DV] (15) D"= DD' + [DV(D'^2 - 1) / D'V']
Therefore with (12) and (15):
16) D"- DD' = D'V'(D^2 - 1) / DV = DV(D'^2 - 1) / D'V'
C- This last equation allows us to define a constant parameter K as:
(17) K=D^2. V^2 / (D^2 -1) = D'^2.V'^2 / (D'^2 - 1)
The parameter K has the same value for two arbitrary speeds ( with their respective D). It is therefore the same for all speeds. In addition, V = 0 gives x = x' and t = t', therefore D = 1 in (8), we must choose the positive solution of equation (17)
(18) D= 1/ sqrt(1-V^2/K)
That’s the famous relation we were looking for… Plugging this in equation (8), gives the transformation (x, t)®(x', t). Poincaré generalises this easily to the general transformation (x, y, z, t) ® (x', y', z', t').
(19) x'=(x-Vt) /sqrt(1-V^2/K) ; y'=y ; z'=z t'=[(t-(Vx/K)] / sqrt(1-V^2/K)
K looks to be a free parameter giving the Galilée transformation if infinite and Lorentz transformation if K = c^2. Obviously these transformations are very close when V/c << 1. Parameter K can’t be negative (it should be possible to travel backwards in time) and it square root looks to be a maximum speed for V. This is confirmed by the square root (1-V^2/K) and also by the speed combinaison equation deducted from (12) and (13): (19) V" = (V+V')/ [1 + (VV'/ K)] if we pose sqrt(K) = k (19’) (k-V")/(k+V") = [(k-V)/(k+V)].[(k-V')/(k+v')] therefore êV ê and êV' ê < k imply êV" ê < k. Poincaré and Lorentz obviously selected K = c^2, in accordance with light speed invariance and conservation of the Maxwell equations in inertial frames. Annexe 2 : Einstein method ( Zur Elektrodynamik der bewegten Körper. Annalen der Physik.) Einstein first states the two ( independent and non contradictory) postulates of the Special Relativity. - Relativity principle: all the physical laws are the same in all inertial frames. - Speed of light is the same in all inertial frame and is equal to “c” Then, Einstein analyses the measurement methods for deriving his equations. We use the original Einstein notations but equations have been labeled He defines how to synchronise all the clocks of a frame ( Poincaré method) with obvious notation ( Signal emitted in A at T0, received and reflected at B at T1, received back in A at T2) T1 = 1/2(T0+T2). (1) This measurement is performed in a frame (R1) sliding at constant speed "v " along "x" axis of frame R0 ( supposed here at rest, arbitrary choice). Let’s try to find the value of « T1 » in frame R0 . For a master ruler of length L in R1, it is easy to show that in R0 ( coordinate "t" ), we have: t1-t0 = L/(V-v) et t2 - t1 = L/(V+v) with V = light speed. (2) . Let introduce x' = x-vt, which maps a point at rest in R1 (coordinates in greek letters) to a fix point of R0 ( coordinates in latin letters). Let’s write Equation (1) in R1 1/2(t0 + t2) = t1 (3) As t= t(x,y,z,t) , with the relevant values of the arguments we get (4) 1/2 [t (0,0,0,t) + t (0,0,0,{t +x'/(V-v) + x'/(V+v)})] = t (x',0,0,t+x'/(V-v)) (5) if x' is supposed to be infinitesimal, we get: (1/2)dt (2,0)= dt (1) that is (1/2) (¶t/¶t) dt = (¶t/¶x'). dx' + (¶t/¶t) . dt (6) Applying to (5) and plugging values of dt in the two members of the equation dt = dx' [( 1/(V-v) + 1 /(V+v)] on the left, and dt = dx' ( 1/(V-v)) on the right , we get: 1/2 ( [1/(V-v)] + [1/(V+v)] )¶t/¶t = (¶t/¶x') + [1/(V-v)]. ¶t/¶t that is : ¶t/¶x' + [v/(V²-v²)] ¶t/¶t = 0 (7) Equation easy to solve whether we suppose that t is a linear function of t,x' t = a ( t - v.x’/(V²-v²)) (8) where a is a function of v, Now let’s replace x' by its value ( x-vt) in (8) and et let’s simplify, we get t = j (v).b.( t - vx/V²) (9) where j(v) is a function of v with b = (1- v²/V²)-1/2 (10)
Let’s apply equation (9) first for + v and then for -v ( we should the initial coordinates again ) to find : j (v). j (-v) = j² (v) = 1: j (v) = j (-v) according to the symmetry of the problem. (11) therefore t = b.( t - vx/V²) (12) Computing space coordinate x from t ( corresponding to à x) is an easy game, just apply the fact the the speed of light is constant x = Vt ( we know t ). The other space coordinates are identical in both frames, in our example. One can check constancy of speed light by x²+y²+z² = V²t² = V²t² =x²+h²+z² The following part of the article is related to the general equations of Lorentz, the metric and the Maxwell equation invariance under these transformations. |